Integral representation theorem for classical Wiener space

In mathematics, the integral representation theorem for classical Wiener space is a result in the fields of measure theory and stochastic analysis. Essentially, it shows how to decompose a function on classical Wiener space into the sum of its expected value and an Itô integral.

Statement of the theorem

Let $C_0([0,T];\mathbb{ℝ})$ (or simply $C_0$ for short) be classical Wiener space with classical Wiener measure $\gamma$. If $F\in L^2(C_0;\mathbb{ℝ})$, then there exists a unique Itô integrable process ${\alpha }^F:[0,T]\times C_0\to \mathbb{ℝ}$ (i.e. in $L^2(B)$, where $B$ is canonical Brownian motion) such that

$F(\sigma )=\underset{C_0}{\int }F(p)\mathrm{d}\gamma (p)+\underset{0}{\overset{T}{\int }}{\alpha }^F(\sigma {)}_t\mathrm{d}{\sigma }_t$

for $\gamma$-almost all $\sigma \in C_0$.

In the above,

• $\underset{C_0}{\int }F(p)\mathrm{d}\gamma (p)=\mathbb{𝔼}[F]$ is the expected value of $F$; and
• the integral $\underset{0}{\overset{T}{\int }}\cdots \mathrm{d}{\sigma }_t$ is an Itô integral.

The proof of the integral representation theorem requires the Clark-Ocone theorem from the Malliavin calculus.

Corollary: integral representation for an arbitrary probability space

Let $(\mathrm{\Omega },{ℱ},\mathbb{\mathbb{P}})$ be a probability space. Let $B:[0,T]\times \mathrm{\Omega }\to \mathbb{ℝ}$ be a Brownian motion (i.e. a stochastic process whose law is Wiener measure). Let $\{{{ℱ}}_t|0\le t\le T\}$ be the natural filtration of ${ℱ}$ by the Brownian motion $B$:

${{ℱ}}_t=\sigma \{B_s^{-1}(A)|A\in \mathrm{B}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}(\mathbb{ℝ}),0\le s\le t\}.$

Suppose that $f\in L^2(\mathrm{\Omega };\mathbb{ℝ})$ is ${{ℱ}}_T$-measurable. Then there is a unique Itô integrable process $a^f\in L^2(B)$ such that

$f=\mathbb{𝔼}[f]+\underset{0}{\overset{T}{\int }}{a}_t^f\mathrm{d}{B}_t$ $\mathbb{\mathbb{P}}$-almost surely.

References

• Mao Xuerong. Stochastic differential equations and their applications. Chichester: Horwood. (1997)